Makoto Suwama - cp's OEIS frontend

User: Makoto Suwama

Makoto Suwama has authored 1 sequences.

A336762 Integers b where the number of cycles under iteration of sum of squares of digits in base b is exactly two.

6, 10, 16, 20, 26, 40, 8626, 481360

Offset: 1

Makoto Suwama, Aug 03 2020

Let b > 1 be an integer, and write the base b expansion of any nonnegative integer n as n = x_0 + x_1 b + ... + x_d b^d with x_d > 0 and 0 <= x_i < b for i=0,...,d.
Consider the map S_{x^2,b}: N to N, with S_{x^2,b}(n) := x_0^2+ ... + x_d^2.
It is known that the orbit set {n, S_{x^2,b}(n), S_{x^2,b}(S_{x^2,b}(n)), ...} is finite for all n > 0. Each orbit contains a finite cycle, and for a given b, the union of such cycles over all orbit sets is finite. Let us denote by L(x^2,i) the set of bases b such that the set of cycles associated to S_{x^2,b} consists of exactly i elements. In this notation, the sequence is the set of known elements of L(x^2,2).
A 1978 conjecture of Hasse and Prichett describes the set L(x^2,2). New elements have been added to this set in the paper Integer Dynamics, by D. Lorenzini, M. Melistas, A. Suresh, M. Suwama, and H. Wang. The sequence contains all b <= 10^6 that are in L(x^2,2).

			For instance, b = 10 is in this sequence since in the decimal system, there are exactly two cycles (1) and (4, 16, 37, 58, 89, 145, 42, 20).

H. Hasse and G. Prichett, A conjecture on digital cycles, J. reine angew. Math. 298 (1978), 8--15. Also on GDZ.
D. Lorenzini, M. Melistas, A. Suresh, M. Suwama, and H. Wang, Integer Dynamics, preprint.

Cf. A193583 and A193585 (b is in this sequence if A193583(b)+A193585(b) = 2).
Cf. A336744 (3 cycles).
Cf. A336783 (4 cycles with sum of cubes of the digits).

cp's OEIS Frontend

User: Makoto Suwama

A336762 Integers b where the number of cycles under iteration of sum of squares of digits in base b is exactly two.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs